cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A145158 G.f. A(x) satisfies A(x/A(x)^2) = 1/(1-x).

Original entry on oeis.org

1, 1, 3, 16, 121, 1143, 12570, 154551, 2072547, 29829412, 455731327, 7332989616, 123548350018, 2169987439342, 39595583375433, 748541216196285, 14628467191450947, 294984129900772611, 6128372452917891216
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 16*x^3 + 121*x^4 + 1143*x^5 +...
x/A(x)^2 = x - 2*x^2 - 3*x^3 - 18*x^4 - 150*x^5 - 1518*x^6 -...
1/A(x) = 1 - x - 2*x^2 - 11*x^3 - 88*x^4 - 869*x^5 - 9876*x^6 -...
Series_Reversion[x/A(x)^2] = x + 2*x^2 + 11*x^3 + 88*x^4 + 869*x^5 +...

Links

Crossrefs

Cf. A145161 (A^3); A088713, A145160, A145162, A145165, A145167.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,B=serreverse(x/A^2);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^2.
Self-convolution yields A145159.

A145160 G.f. A(x) satisfies A(x/A(x)^3) = 1/(1-x).

Original entry on oeis.org

1, 1, 4, 31, 347, 4860, 79174, 1440837, 28584939, 608533714, 13751688892, 327333165775, 8160149459870, 212121519165566, 5730205766494409, 160425928432680795, 4644491031188023566, 138792548776938444503
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Cf. A145161 (A^3); A088713, A145158, A145162, A145165, A145167.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(n=0,n,B=serreverse(x/A^3);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^3.
Self-convolution cube yields A145161.

A145162 G.f. A(x) satisfies A(x/A(x)^4) = 1/(1-x).

Original entry on oeis.org

1, 1, 5, 51, 757, 14058, 303443, 7313188, 192096189, 5413972155, 161972306602, 5104569475976, 168500227127871, 5800706769824992, 207552636468976072, 7697809237540240440, 295284422299359774761, 11693774821978063710405
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Cf. A145163 (A^2), A145164 (A^4); A088713, A145158, A145160, A145165, A145167.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^4);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^4.
Self-convolution square yields A145163.
Self-convolution 4th power yields A145164.

A145165 G.f. A(x) satisfies A(x/A(x)^5) = 1/(1-x).

Original entry on oeis.org

1, 1, 6, 76, 1406, 32531, 874407, 26234503, 857727024, 30087607090, 1120358453641, 43948073274103, 1805827523343241, 77390779901965470, 3447553371343457810, 159209478315871014816, 7605143367385966288569
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Cf. A145166 (A^5); A088713, A145158, A145160, A145162, A145167.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^5);A=1/(1-B));polcoeff(A,n)}

Formula

G.f. satisfies: 1 - 1/A(x) = x*A( 1 - 1/A(x) )^5.
Self-convolution 5th power yields A145166.

A145169 G.f. A(x) satisfies A(x/A(x)^2) = 1/(1-x)^3.

Original entry on oeis.org

1, 3, 24, 361, 7851, 214245, 6815986, 242659191, 9431036685, 393771017068, 17469277612998, 817033067908275, 40050828368606412, 2048642917085523633, 108968676087557745858, 6010617981960195923285, 343038687551360605111371
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Links

Crossrefs

Cf. A145167, A145168, A145170 (A^2).

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^2);A=1/(1-B)^3);polcoeff(A,n)}

Formula

Self-convolution cube of A145167.
Self-convolution square yields A145170.

A145170 G.f. A(x) satisfies A(x/A(x)) = 1/(1-x)^6.

Original entry on oeis.org

1, 6, 57, 866, 18444, 492924, 15424611, 542166480, 20861518935, 864061112296, 38081996557383, 1771322835258594, 86425203984341130, 4402953230795279532, 233372023965531945057, 12832558973488295874402, 730347857708249147767893
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Links

Crossrefs

Cf. A145167, A145168, A145169.

Programs

  • Maple
    A[0]:= x -> 1+c*x:
for n from 1 to 20 do
  cc:= coeff(series(A[n-1](x/A[n-1](x))-1/(1-x)^6, x, n+1),x,n);
  A[n]:= unapply(eval(A[n-1](x),c=solve(cc,c))+c*x^(n+1),x);
od:
seq(coeff(A[20](x),x,j),j=0..20); # Robert Israel, Aug 19 2018
  • Mathematica
    nmax = 16; sol = {a[0] -> 1};
Do[A[x_] = Sum[a[k] x^k, {k, 0, n}] /. sol; eq = CoefficientList[A[x/A[x]] - 1/(1 - x)^6 + O[x]^(n + 1), x] == 0 /. sol; sol = sol ~Join~ Solve[eq] [[1]], {n, 1, nmax}]; sol /. Rule -> Set;
a /@ Range[0, nmax] (* Jean-François Alcover, Nov 01 2019 *)
  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A);A=1/(1-B)^6);polcoeff(A,n)}

Formula

Self-convolution 6th power of A145167.
Self-convolution cube of A145168.
Self-convolution square of A145169.

A145168 G.f. A(x) satisfies A(x/A(x)^3) = 1/(1-x)^2.

Original entry on oeis.org

1, 2, 15, 226, 4959, 136338, 4361818, 155947386, 6080986701, 254568183028, 11318103523886, 530301559146450, 26035289254487166, 1333502730683975402, 71012366873757289557, 3921010145962481940986, 223985973308662486807733
Offset: 0

Views

Author

Paul D. Hanna, Oct 03 2008

Keywords

Crossrefs

Cf. A145167, A145169, A145170 (A^3).

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n),B);for(n=0,n,B=serreverse(x/A^3);A=1/(1-B)^2);polcoeff(A,n)}

Formula

Self-convolution square of A145167.
Self-convolution cube yields A145170.
Showing 1-7 of 7 results.

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